Question

When two chords of a circle bisect each other, then which of the following statements is true?

• A. Both chords are perpendicular to each other.
• B. Both chords are parallel to each other.
• C. Both chords are unequal.
• D. Both are diameters of the circle.

Answer 1

The correct option is D

Both are diameters of the circle.

Step-1 Explanation for correct option:

(D) Both are diameters of the circle.

Diameters are the only chords of a circle that passes through the centre of a circle and they bisect every other chord inside the circle that they intersect with.

Hence when two chords of a circle bisect each other they are two diameters of the circle.

Step-2 : Explanation for incorrect options:

(A) When two chords of a circle bisect each other, then they need not be mutually perpendicular.

Let $AB,CD$ be the chords intersecting at P such that $AP=PB,CP=PD$. As $PC.PD=PA.PB$, we get $P{A}^2=P{C}^{2}orPA=PC$. This means $AB=CD$, i.e., the chords are of equal length. The perpendicular to $AB$ and $CD$ at their midpoints pass through the centre of the circle.

Unless $P$ is the centre of the circle this cannot happen. Thus$AB$ and $CD$ are diameters of the circle. They need not be mutually perpendicular.

(B) When two chords of a circle bisect each other, then Both chords cannot be parallel to each other.

(C) When two chords of a circle bisect each other, then they are two diameters of the circle Both chords cannot be unequal.

Consider the above diagram $AB,CD$ be the chords intersecting at P such that $AP=PB,CP=PD$. As $PC.PD=PA.PB$, we get $P{A}^2=P{C}^{2}orPA=PC$. This means $AB=CD$, i.e., the chords are of equal length.

Hence, option (D) is the correct answer.